Hilbert’s MistakeEdward NelsonDepartment of MathematicsPrinceton University1AbstractHilbert was at heart a Platonist. (“No oneshall expel us from the paradise that Cantor hascreated for us.”) His formalism was primarily atactic in his battle against Brouwer’s intuitionism.His mistake was to pose the problem ofshowing that mathematics, beginning with PeanoArithmetic, is consistent, rather than to ask whetherit is consistent.In this talk I give reasons for taking seriouslythe possibility that contemporary mathematics,including Peano Arithmetic, may indeed be incon-sistent.2Potential vs. Completed InfinityLet us distinguish between the genetic, in thedictionary sense of pertaining to origins, and theformal. Numerals (terms containing only the unaryfunction symbol S and the constant 0) are genetic;they are formed by human activity. All of mathe-matical activity is genetic, though the subject mat-ter is formal.Numerals constitute a potential infinity. Givenany numeral, we can construct a new numeral byprefixing it with S.Now imagine this potential infinity to be com-pleted. Imagine the inexhaustible process of con-structing numerals somehow to have been finished,and call the result the set of all numbers, denotedby N.Thus N is thought to be an actual infinity ora completed infinity. This is curious terminology,since the etymology of “infinite” is “not finished”.3We were warned.Aristotle: Infinity is always potential, neveractual.Gauss: I protest against the use of infinitemagnitude as something completed, which is neverpermissible in mathematics.We ignored the warnings.With the work of Dedekind, Peano, and Can-tor above all, completed infinity was accepted intomainstream mathematics.Mathematics became a faith-based initiative.Try to imagine N as if it were real.A friend of mine came across the following onthe Web:4www.completedinfinity.comBuy a copy of N!Contains zero—contains the successor of ev-erything it contains—contains only these.Just $100.Do the math! What is the price per number?Satisfaction guaranteed!Use our secure form to enter your credit card numberand its security number, zip code, social security number,bank’s routing number, checking account number, date ofbirth, and mother’s maiden name.The product will be shipped to you withintwo business days in a plain wrapper.5My friend answered this ad and proudly showedhis copy of N to me. I noticed that zero was green,and that the successor of eve ry green number wasgreen, but that his model contained a red number.I told my friend that he had been cheated, and hadbought a nonstandard model, but he is color blindand could not see my point.I bought a model from another dealer and amquite pleased with it. My friend maintains thatit contains an ineffable number, although zero iseffable and the successor of every effable number iseffable, but I don’t know what he is talking about.I think he is just jealous.The point of this conceit is that it is impos-sible to characterize N unambiguously, as we shallargue in detail.6As a genetic concept, the notion of numeral isclear. The attempt to formalize the concept usu-ally proceeds as follows:(i) zero is a number;(ii) the successor of a number is a number;(iii) zero is not the successor of any number;(iv) different numbers have different successors;(v) something is a number only if it is so byvirtue of (i) and (ii).We shall refer to this as the usual definition.Sometimes (iii) and (iv) are not stated explicitly,but it is the extremal clause (v) that is unclear.What is the meaning of “by virtue of”? It isobviously circular to define a number as somethingconstructible by applying (i) and (ii) any numberof times.We cannot characterize numbers from below,so we attempt to characterize them from above.7The study of the foundations of arithmeticbegan in earnest with the work of Dedekind andPeano. Both of these authors gave what todaywould be called set-theoretic foundations for arith-metic. In ZFC (and extensions by definitions thereof)let us write 0 for the empty set and define the suc-cessor bySx = x ∪ { x }We definex is inductive ↔ 0 ∈ x & ∀y [ y ∈ x→ Sy ∈ x ].Then the axiom of infinity of ZFC is∃x [ x is inductive ]and one easily proves in ZFC that there exists aunique smallest inductive set; i.e.,∃!xx is inductive & ∀y [ y is inductive→ x ⊆ y ] 8We define the constant N to be this smallestinductive set:N = x ↔ x is inductive & ∀y [ y isinductive → x ⊆ y ]and we definex is a number ↔ x ∈ N.Then the following are theorems:(1) 0 is a number(2) x is a number → Sx is a number(3) x is a number → Sx 6= 0(4) x is a number & y is a number & x 6= y→ Sx 6= Sy.9These theorems are a direct expression of(i)–(iv) of the usual definition. But can we expressthe extremal clause (v)? The induction theoremx is a number & y is inductive → x ∈ ymerely asserts that for any property that can beexpressed in ZFC, if 0 has the property, and if thesuccessor of every element that has the propertyalso has the property, then every number has theproperty.We cannot say, “For all numbers x there ex-ists a numeral d such that x = d” since this is acategory mistake conflating the formal with thegenetic.10Using all the power of modern mathematics,let us try to formalize the concept of number.Let T be any theory whose language containsthe constant 0, the unary function symbol S, andthe unary predicate symbol “is a number”, suchthat (1)–(4) are theorems of T.For example, T could be the extension by defi-nitions of ZFC described above or it could be PeanoArithmetic P with the definition:x is a number ↔ x = x.Have we captured the intended meaning of theextremal clause (v)?To study this question, construct Tϕby ad-joining a new unary predicate symbol ϕ and theaxioms(5) ϕ(0)(6) ϕ(x) → ϕ(Sx).11Notice that ϕ is an undefined symbol.If T is ZFC, we cannot form the set{ x ∈ N : ϕ(x) } because the subset axioms of ZFCrefer only to formulas of ZFC and ϕ(x) is not s ucha formula. Sets are not genetic objects, and to askwhether a set with a certain property exists is toask whether a certain formula beginning with ∃can be proved in the theory.Similarly, if T is P we cannot apply inductionto ϕ(x) since this is not a formula of P. Inductionis not a truth; it is an axiom scheme of a formaltheory.If T is consistent then so is Tϕ, because wecan interpret ϕ(x) by x = x. (And conversely,of course, if T is inconsistent then so is Tϕ.) Forany